IGAP/MPIM lecture course "From 3-manifold invariants to number theory" - Don Zagier

[ICTP Math Section]

Lecturer: Don Zagier
Course Type: PhD Course
Academic Year: 2020-2021
Period: Spring 2021
Duration:  40 h

http://indico.ictp.it/event/9681/

Research Group: Geometry and Mathematical Physics
Location: Online/Presence (this will be a hybrid Lecture course)

Description:

Questions from topology have led to interesting number theory for many 
years, a famous example being the occurrence of Bernoulli numbers in 
connection with stable homotopy groups and exotic spheres, but some 
developments from the last few years have led to much deeper 
relationships and to highly non-trivial ideas in number theory.  The 
course will attempt to describe some of these new interrelationships, 
which arise from the study of quantum invariants of knot complements and 
other 3-dimensional manifolds. [Joint work with Stavros Garoufalidis]

Topics to be studied include:

* The dilogarithm function, the 5-term relation, and triangulations of 
3-manifolds

* Quantum invariants of 3-folds (Witten-Reshetikhin-Turaev and Kashaev 
invariant) definitions and first properties

* The Habiro ring (this is a really beautiful algebraic object that 
should be much better known and in which both of the above-named quantum 
invariants live)

* Perturbative series (formal power series in h) associated to knots

* Turning divergent power series into actual functions (this has 
connections with resurgence theory and involves some quite fun analytic 
considerations)

* Numerical methods (the ones needed are surprisingly subtle)

* Holomorphic functions in the upper half-plane (q-series) associated to 
knots

* Modular properties of both the Habiro-like and of the holomorphic 
invariants

These topics are all interconnected in a very beautiful way, formally 
summarized at the end by a single matrix invariant having different 
realizations in the Habiro world, the formal power series world, and the 
q-series world.

Although some quite advanced topics will be reached or touched upon, the 
course assumes no prerequisites beyond standard basic definitions from 
either topology, number theory, or analysis.

To access the course use the following Zoom coordinates:

Zoom meeting ID 969 5251 6566
Password  307018
ZoomLink: https://zoom.us/j/96952516566?pwd=Z3NyZW04M2YxSHo2MWdlOHJ4MlNpUT09

Next Lectures:

Friday, June 18, 2021 - 14:00 - 16:00 (Euler Lecture Hall)

Monday, June 21, 2021 - 16:00 - 18:00 (Luigi Stasi Seminar Room)

Friday, June 25, 2021 - 14:00 - 16:00 (Euler Lecture Hall)

Monday, June 28, 2021 - 16:00 - 18:00 (Euler Lecture Hall)

Friday, July 2, 2021 - 14:00 - 16:00 (Euler Lecture Hall)

Monday, July 5, 2021 - 16:00 - 18:00 (Budinich Lecture Hall)

Friday, July 9, 2021 - 14:00 - 16:00 (Budinich Lecture Hall)

Monday, July 12, 2021 - 16:00 - 18:00 (Budinich Lecture Hall)

Friday, July 16, 2021 - 14:00 - 16:00 Budinich Lecture Hall)

-- 
Koutou Mabilo
ICTP Mathematics Group
Leonardo Da Vinci Building
Strada Costiera no. 11
34151 Trieste, Italy
Tel. no.: +39-040-2240455

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